Edinburgh University students could pay £36,000
Edinburgh University could become the most expensive place to study in the UK for students from England, Wales and Northern Ireland.
The Scottish university said it would charge £9,000 a year in tuition fees.
The National Union of Students Scotland said £36,000 for a standard four-year degree at Edinburgh was "both staggering and ridiculous".
Edinburgh is the third Scots university to set its fees for rest of UK (RUK) residents at the maximum level.
However, Aberdeen and Heriot-Watt universities said they intended to cap their fees at a maximum of £27,000 for the course, even if it was four years, the standard length of Scottish courses.
The University of Edinburgh said it would not do this but it would offer "the most generous bursary package within the UK for those on the lowest household incomes".
The Scottish government said a consultation on its proposals for tuition fees for students who come to Scotland to study from other parts of the UK had closed last week.
It said fee levels were being set by Scottish universities on an "indicative basis", pending the outcome of the consultation and the subsequent legislative process.
However, the universities intend to bring in the fees next summer if they get the go-ahead.
Students who are resident in Scotland do not pay tuition fees at Scottish universities.
The SNP made a pledge before it was elected in May that it would not introduce fees or graduate contributions from students who live in Scotland.
In June, Education Secretary Mike Russell outlined proposals to allow Scottish universities to set their own fees for UK residents from England, Wales and Northern Ireland.
The move was prompted by the UK government's decision to raise the cap on fees in the rest of the UK to £9,000.
Mr Russell said Scottish universities would be free to set fees of between the current £1,800 and £9,000 but said he expected levels to be lower than those south of the border.
Responding to the Edinburgh and Heriot-Watt announcements, a Scottish Government spokesman said: "It is up to individual universities to manage and set their fee levels for students from the rest of the UK, bearing in mind the need to be competitive and attractive to a broad range of students.
"It is in their interests, as well as Scotland's, that we maintain the cosmopolitan character of our student population at the same time as making sure that opportunities for students who live in Scotland are protected."
Heriot-Watt said it wanted to cap fees for English, Welsh and Northern Ireland students at £27,000, even if they study for more than three years.
The university said it would encourage as many students as possible to skip their first year of study if their A-level results enabled them to do so, while waiving the first year's fee for students who opted to take a four-year course.
The National Union of Students in Scotland said Heriot-Watt's decision was "terrible".
After the Edinburgh decision, Graeme Kirkpatrick, depute president of NUS Scotland, said: "The average cost to study at Oxford and Cambridge is around £25,000 in fees, which while still eye-wateringly large, pales in comparison with this.
"And that's before you add additional debt for the extra year of living costs for the four-year degree in Scotland.
"This is nothing less than cashing in on students from the rest of the UK, and giving the signal that Edinburgh University is more interested in the money you can bring, as opposed to your academic ability.
"The reputational damage this could do, not only to Edinburgh but to the whole of Scottish higher education, should not be underestimated."
Prof Mary Bownes, vice principal for external engagement at the University of Edinburgh, said: "More than 50% of additional tuition fee income will go towards bursaries with the remainder going towards enhancing the student experience.
"There will also be a major new internally-funded scheme for access and accommodation bursaries for Scottish domiciled students."
Professor Bownes added: "The increase in the fee is necessary as we will no longer receive government funding for RUK (rest of United Kingdom)-domiciled students.
"These students will be studying at one of the world's top teaching and research institutions, regularly ranked amongst the leading universities in the world."
However, Mary Senior, Scottish official of the University and College Union, said: "We are disappointed that Heriot-Watt and Edinburgh universities are to set headline fees at the highest level, leaving students paying more to study in Scotland than even in England.
"Though Edinburgh are offering bursaries most students won't qualify for these so that only the wealthiest will consider Edinburgh an option.
"Our worst fears that most Scottish universities will ignore the minister and charge the highest possible fee have come to fruition."
Last month, Prof Iain Diamond, principal of the University of Aberdeen, said that maintaining fees at the current level - about £1,800 a year for most degrees - was no longer possible.